The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables

Zelený, Miroslav

Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 405-428

We construct a differentiable function $f:{\mathbf{R}}^n\to \mathbf{R}$ ($n\ge 2$) such that the set ${\left(\nabla f\right)}^{-1}\left(B(0,1)\right)$ is a nonempty set of Hausdorff dimension $1$. This answers a question posed by Z. Buczolich.

On construit une fonction différentiable $f:{\mathbf{R}}^n\to \mathbf{R}$ ($n\ge 2$) telle que l’ensemble ${\left(\nabla f\right)}^{-1}\left(B(0,1)\right)$ est non vide et sa dimension de Hausdorff est $1$. C’est une réponse à une question posée par Z. Buczolich.

Classification : 26B05, 28A75
Keywords: Denjoy–Clarkson property, gradient, Hausdorff measure, infinite game
Mots-clés : propriété de Denjoy-Clarkson, gradient, mesure de Hausdorff, jeu infini

[1] Bruckner, A. Differentiation of real functions, CRM Monograph Series, 5, American Mathematical Society, Providence, RI, 1994 | Zbl | MR

[2] Buczolich, Z. The $n$-dimensional gradient has the 1-dimensional Denjoy-Clarkson property, Real Anal. Exchange, Volume 18 (1992-93), pp. 221-224 | Zbl | MR

[3] Buczolich, Z. Another note on the gradient problem of C. E. Weil, Real Anal. Exchange, Volume 22 (1996-97), pp. 775-784 | Zbl | MR

[4] Buczolich, Z. Solution to the gradient problem of C. E. Weil, Rev. Mat. Iberoamericana, Volume 21 (2005), pp. 889-910 | MR | EuDML | Zbl

[5] Clarkson, J. A. A property of derivatives, Bull. Amer. Math. Soc., Volume 53 (1947), pp. 124-125 | DOI | Zbl | MR

[6] Denjoy, A. Sur une proprieté des fonctions dérivées, Enseignement Math., Volume 18 (1916), pp. 320-328 | JFM

[7] Deville, R.; Matheron, É. Infinite games, Banach space geometry and the eikonal equation, Proc. Lond. Math. Soc. (3), Volume 95 (2007) no. 1, pp. 49-68 | DOI | Zbl | MR

[8] Engelking, R. General Topology, Heldermann Verlag, Berlin, 1989 | Zbl | MR

[9] Holický, P.; Malý, J.; Weil, C. E.; Zajíček, L. A note on the gradient problem, Real Anal. Exchange, Volume 22 (1996-97), pp. 225-235 | Zbl | MR

[10] Malý, J. The Darboux property for gradients, Real Anal. Exchange, Volume 22 (1996/97) no. 1, pp. 167-173 | Zbl | MR

[11] Malý, J.; Zelený, M. A note on Buczolich’s solution of the Weil gradient problem: a construction based on an infinite game, Acta Math. Hungar., Volume 113 (2006), pp. 145-158 | DOI | Zbl

[12] Mattila, P. Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, Volume 44, Cambridge University Press, Cambridge, 1995 | Zbl | MR

[13] Weil, C. E. Query 1, Real Anal. Exchange, Volume 16 (1990-91), pp. 373